It is one
thing to understand the concept of addition and subtraction of whole numbers
and integers.Keeping track of the
complexities of these operations when working with large numbers is a
different matter.For that, we need a
concept model to illustrate the place value system of numeration and an
expanded algorithm to keep track of details.Figure 2.8 shows two concept models for the operation 109 + 211 =
320.Both models are based on base ten
blocks.One model is very literal,
showing in detail the individual ones in the tens and hundreds graphics.The other is more symbolic, without
proportional scaling between the graphical icons representing ones, tens, and
hundreds.Note that in each representation,
10 ones are regrouped as 1 ten.

Figure
2.9 shows pairs the second of these concept models with an expanded algorithm
for the same operation.The graphical and
numerical representations both contain essentially the same information.In discussions based on the Concept –
Written Notations– Spoken Language model shown in Figure 1.3, students should
develop fluency in representing and explaining arithmetic operations with
whole numbers.This is the first step
in developing a genuine understanding of the logical bases for arithmetic
operations.

3(100)

2(10)

0(1)

Hundreds

Tens

Ones

Concept Model:
Base Ten Blocks

Algorithm: The expanded algorithm is a numerical
representation that helps to clarify the meaning of place value.

1(10)

1(100) + 0(10) +9(1)

+2(100) + 1(10) +1(1)

3(100) + 2(10) +10(1)

3(100) + 2(10)
+ 0(1)

Written
in standard form, the answer as 320

Figure 2.9: Concept Model and Expanded Algorithm
for 109 + 211 = 320

Figure
2.10 demonstrates the use of this approach in the context of a related
subtraction problem involving regrouping, 320 – 109 = 211.

2(100)

1(10)

1(1)

Hundreds

Tens

Ones

Concept Model:
Base Ten Blocks

Algorithm: The expanded algorithm is a numerical
representation that helps to keep place value.

1(10)10(1)

3(100) + 2(10) +0(1)

–
1(100) – 0(10) –
9(1)

2(100) + 1(10)+1(1)

Written
in standard form, the answer as 211

Figure 2.10: Concept Model and Expanded Algorithm
for 320 – 109 = 211

Figure
2.11 shows the “standard” algorithms associated with the operations modeled
in Figures 2.9 and 2.10.These
notations provide little conceptual support for learners.They were designed as procedural notations
for use by people who already
understand the conceptual basis for computation but who must also become
reliable and fast in routine computation.Unfortunately for young learners, the very brevity of these notations
is often a source of genuine confusion.For instance, in the process of “borrowing”, a 1 is written above the
ones and tens columns of the subtraction problem.These 1’s are identical in appearance but
represent different quantities.Furthermore, the oral language used to describe arithmetic operations
is often misleading.Referring to the
addition problem in Figure 2.11, a well-meaning adult might say something
like “Nine plus one equals ten … write the zero … carry the one … one plus
one equals two ...”, obscuring the fact that 10 ones are regrouped into 1 ten
and that ten is relocated in the tens column.Statements of this sort hide the meaning of regrouping.

Developing
proficiency in the use of standard algorithms is a goal of elementary
mathematics education.But the process
by which students acquire this skill should have as its foundation a deep
understanding of the meaning of arithmetic algorithms.The use of concept models and expanded
algorithms is a powerful tool in the development of this foundation.